Journal of Complex Analysis

Volume 2014 (2014), Article ID 864805, 5 pages

http://dx.doi.org/10.1155/2014/864805

Research Article

A New Class of Meromorphic Functions Involving the Polylogarithm Function

Khadeejah Rasheed Alhindi and Maslina Darus

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia (UKM), 43600 Bangi, Selangor Darul Ehsan, Malaysia

Received 10 May 2014; Revised 25 July 2014; Accepted 26 July 2014; Published 11 August 2014

Academic Editor: Jacek Dziok

Copyright © 2014 Khadeejah Rasheed Alhindi and Maslina Darus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We introduce a new operator associated with polylogarithm function. By making use of the new operator, we define a certain new class of meromorphic functions and discussed some important properties of it.

1. Introduction

Historically, the classical polylogarithm function was invented in 1696, by Leibniz and Bernoulli, as mentioned in [1]. For and a natural number with , the polylogarithm function (which is also known as Jonquiere’s function) is defined by the absolutely convergent series:

Later on, many mathematicians studied the polylogarithm function such as Euler, Spence, Abel, Lobachevsky, Rogers, Ramanujan, and many others [2], where they discovered many functional identities by using polylogarithm function. However, the work employing polylogarithm has been stopped many decades later. During the past four decades, the work using polylogarithm has again been intensified vividly due to its importance in many fields of mathematics, such as complex analysis, algebra, geometry, topology, and mathematical physics (quantum field theory) [3–5]. In 1996, Ponnusamy and Sabapathy discussed the geometric mapping properties of the generalized polylogarithm [6]. Recently, Al-Shaqsi and Darus generalized Ruscheweyh and Salagean operators, using polylogarithm functions on class of analytic functions in the open unit disk . By making use of the generalized operator they introduced certain new subclasses of and investigated many related properties [7]. A year later, same authors again employed the th order polylogarithm function to define a multiplier transformation on the class in [8].

To the best of our knowledge, no research work has discussed the polylogarithm function in conjunction with meromorphic functions. Thus, in this present paper, we redefine the polylogarithm function to be on meromorphic type.

Let denote the class of functions of the form

which are analytic in the punctured open unit disk

A function in is said to be meromorphically starlike of order if and only if

for some . We denote by the class of all meromorphically starlike order . Furthermore, a function in is said to be meromorphically convex of order if and only if

for some . We denote by the class of all meromorphically convex order . For functions given by (2) and given by

we define the Hadamard product (or convolution) of and by

Let be the class of functions of the form

which are analytic and univalent in .

Liu and Srivastava [9] defined a function by multiplying the well-known generalized hypergeometric function with as follows:

where are complex parameters and , .

Analogous to Liu and Srivastava work [9] and corresponding to a function given by

we consider a linear operator which is defined by the following Hadamard product (or convolution):

Next, we define the linear operator as follows:

Now, by making use of the operator , we define a new subclass of functions in as follows.

Definition 1. For and , let denote a subclass of consisting functions of form (2) satisfying the condition that

where is given by (12). Furthermore, we say that a function , whenever is of form (8).

In the following sections, we investigate coefficient inequalities, extreme points, radii of starlikeness and convexity of order , and integral means inequalities for the new class .

2. Coefficient Inequalities

The following theorem gives a necessary and sufficient condition for a function to be in the class .

Theorem 2. Let given by (8). Then if and only if

Proof. Suppose that . Then

If we choose to be real and letting , we get

which is equivalent to (14). Conversely, let us suppose that assertion (14) holds true.

Then we can write

Hence, . Finally, we note that inequality (14) is sharp; the extremal function is

3. Extreme Points

In this section, we determine the extreme points for functions in the class .

Theorem 3. Let and

Then, if and only if it can be represented in the form

Proof. Let , , , . Then, we have

Therefore,

Hence, by Theorem 2, .

Conversely, we suppose that , since

We set

and . Then we have

The results follow.

4. Radii of Meromorphic Starlikeness and Meromorphic Convexity

Theorem 4. Let . Then is meromorphically starlike of order in the disk , where

The result is sharp for the extremal function given by (19).

Proof. It is sufficient to show that

which easily follows from (4), since

Considering that , the above expression is less than if and only if

or

By Theorem 2, we have

then, (27) holds true if

which is equivalent to

which yields the starlikeness of the family and completes the proof.

Theorem 5. Let . Then is meromorphically convex of order in the disk , where

The result is sharp for the extremal function given by

Proof. By using the technique employed in the proof of Theorem 4, we can show that

for , and prove that the assertion of the theorem is true.

5. Integral Means Inequalities

Let and be analytic in . Then the function is said to be subordinate to in , written by

if there exists a function which is analytic in with and with and such that for . From the definition of the subordinations, it is easy to show that subordination (37) implies that

Theorem 6 (see [10]). If and are any two functions, analytic in , with , then, for and ; ,

Theorem 7. Suppose and is defined by

If there exists an analytic function such that

then, for and ,

Proof. We need to show that

From Theorem 6, it suffices to prove that

If we set such that

we get

Clearly, ; then from Theorem 2 we can write

That completes the proof.

Conflict of Interests

The authors declare that they have no conflict of interests.

Authors’ Contribution

Both authors read and approved the final paper.

Acknowledgment

The authors would like to thank the center of research and instrumentation (CRIM), National University of Malaysia (UKM), for sponsoring this work under Grant code GUP-2013-004.

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